2025/05/12 by Van Kien, Van Kien, Do, Hop D. Nguyen +1
Computer Science · Mathematics · #13A02 #13F20 #14N05 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2505.07248
openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring (R,\mathfrakm) is Koszul if its residue field R/\mathfrakm has a finite linearity defect. We provide a positive answer to this question when R is a Cohen-Macaulay local ring of almost minimal multiplicity with the residue field of characteristic zero. The proof depends on the study of noetherian local rings (R,\mathfrakm) such that \mathfrakm2 is a principal ideal, which we call g-stretched local rings. The class of g-stretched local rings subsumes stretched artinian local rings studied by Sally, and generic artinian reductions of Cohen-Macaulay local rings of almost minimal multiplicity. An essential part in the proof of our main result is a complete characterization of one-dimensional complete g-stretched local rings. Beside partial progress on Herzog-Iyengar's question, another consequence of our study is a numerical characterization of all g-stretched Koszul rings, strengthening previous work of Avramov, Iyengar, and Şega.